Math Problem Statement

Certainly! Here’s the translation:

"2. Given that one vertex of a rectangle is at (2,1), another vertex is at (6,5), and the length of the diagonal is 4 units, find the coordinates of the remaining two vertices of the rectangle."

Solution

To solve this problem, we start by understanding the given information:

  1. Two vertices of the rectangle are at points A(2,1)A(2, 1) and C(6,5)C(6, 5).
  2. The diagonal length of the rectangle is 4 units.
  3. We need to find the coordinates of the remaining two vertices, say B(x1,y1)B(x_1, y_1) and D(x2,y2)D(x_2, y_2).

Step-by-Step Solution

  1. Diagonal Length Calculation: The length of the diagonal of a rectangle can be found using the distance formula. Given that A(2,1)A(2, 1) and C(6,5)C(6, 5) are two opposite vertices of the rectangle, let's calculate the diagonal length:

    Diagonal length=(62)2+(51)2=42+42=16+16=32=42\text{Diagonal length} = \sqrt{(6 - 2)^2 + (5 - 1)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}

    Since the length of diagonal ACAC is 424\sqrt{2}, but it is given that the diagonal length is 4 units, ACAC cannot be the diagonal. Thus, A(2,1)A(2, 1) and C(6,5)C(6, 5) must be adjacent vertices.

  2. Finding Midpoint and Using Diagonal Properties: For the two unknown vertices BB and DD, let's denote:

    B(x1,y1)andD(x2,y2)B(x_1, y_1) \quad \text{and} \quad D(x_2, y_2)

    Since the diagonals of a rectangle bisect each other, the midpoint of diagonal ACAC is also the midpoint of diagonal BDBD. Let's find the midpoint MM of ACAC:

    M=(2+62,1+52)=(4,3)M = \left( \frac{2 + 6}{2}, \frac{1 + 5}{2} \right) = (4, 3)

    This means that the midpoint of BDBD is also (4,3)(4, 3).

  3. Using Distance to Find Remaining Vertices: We know that the diagonal length BD=4BD = 4. Let’s use the distance formula to express the coordinates of points B(x1,y1)B(x_1, y_1) and D(x2,y2)D(x_2, y_2) in terms of their distance to the midpoint M(4,3)M(4, 3).

    (x14)2+(y13)2=2and(x24)2+(y23)2=2\sqrt{(x_1 - 4)^2 + (y_1 - 3)^2} = 2 \quad \text{and} \quad \sqrt{(x_2 - 4)^2 + (y_2 - 3)^2} = 2

    Squaring both sides:

    (x14)2+(y13)2=4and(x24)2+(y23)2=4(x_1 - 4)^2 + (y_1 - 3)^2 = 4 \quad \text{and} \quad (x_2 - 4)^2 + (y_2 - 3)^2 = 4

    These equations represent two circles of radius 2 centered at M(4,3)M(4, 3).

  4. Determine Specific Coordinates: Since BB and DD are symmetric about point M(4,3)M(4, 3), we can express the solutions as:

    x1=4+a,  x2=4a,  y1=3+b,  y2=3bx_1 = 4 + a, \; x_2 = 4 - a, \; y_1 = 3 + b, \; y_2 = 3 - b

    Substituting into the distance formula gives:

    a2+b2=4a^2 + b^2 = 4

    Choosing a=2,b=0a = 2, b = 0 and a=0,b=2a = 0, b = 2, we find:

    • For (a,b)=(2,0)(a, b) = (2, 0), the points are B(6,3)B(6, 3) and D(2,3)D(2, 3).
    • For (a,b)=(0,2)(a, b) = (0, 2), the points are B(4,5)B(4, 5) and D(4,1)D(4, 1).
  5. Conclusion: The coordinates of the remaining vertices are B(6,3)B(6, 3) and D(2,3)D(2, 3) or B(4,5)B(4, 5) and D(4,1)D(4, 1).

Would you like more details or have any questions?

Related Questions:

  1. How do you find the midpoint of a line segment?
  2. What are the properties of diagonals in a rectangle?
  3. How do you determine if two points are adjacent or opposite vertices of a rectangle?
  4. Can you explain the distance formula and its applications in geometry?
  5. How do you solve for unknown variables using geometric constraints?

Tip:

When working with geometric shapes, visualizing the problem by sketching can often make it easier to understand and solve.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Rectangles

Formulas

Distance formula: √((x2 - x1)^2 + (y2 - y1)^2)
Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)

Theorems

Properties of diagonals in a rectangle
Pythagorean Theorem

Suitable Grade Level

Grades 9-11